Electron microscopy
Linear Least Squares Fitting Technique
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Linear least squares fitting (LLSF) technique is the simplest and most commonly used form of linear regression and provides a solution to a problem of finding the best fitting straight line through a set of points.

There are many applications with this technique, for instance, in the quantifications in electron- and photon-induced X-rays, linear least-square fitting can be used to deconvolute the overlapped peaks once the continuum background is eliminated by a proper technique. In the LLSF method, it is assumed that each channel of the measured spectrum is equal to the sum of a set of the reference spectra in the same channel, as given below,
         rj = Aj1•Rj1 + Aj2•Rj2 + Aj3•Rj3 + ... + Ajn•Rjn + Ej ---------------------- [1760a]
           Linear Least Squares Fitting Technique ---------------------------------- [1760b]
          j -- The channel j.
          rj -- The measured signal in chanel j.
          Aj1, Aj2, Rj3 ... Ajn -- The corresponding coefficients which are used to LLSF fitting.
          Rj1, Rj2, Rj3 ... Rjn -- The signals of the reference spectra in chanel j.
          Ej -- The statistical error which represents the degree of noise and fluctuation in channel j.
          1, 2, ... n -- The pure elements.

The reference spectra may be any set of curves that can be extracted from a proper model, including the information of the shape and amplitude of the particular X-ray lines of interest. In general, such references are a set of measured spectra from the pure elements 1, 2, ... n. According to the least-squares method, the best set of the Ai coefficients are obtained by minimizing the value of x2, which is given by,
        Linear Least Squares Fitting Technique ------------------------------- [1760c]
             Linear Least Squares Fitting Technique ----------------------------- [1760d]
          x2 -- The Chi-Squared.
          σj -- The standard deviation that represents the statistical uncertainty of rj.

In ideal least-squares fitting, if the x2 is normalized (dividing by the number of fitted points minus degrees of freedom), it should be approximately one in average. Values of x2 that are much greater than one indicates some systematic error in the fit.

Different from the ideal case described in Equation 1760b, the practical least-squares fitting must incorporate a treatment of the continuum differences. [1]










[1] McCarthy, J.J., and F.H. Schamber, 1981, Least-Squares Fit with Digital Filter: A Status Report.  In Energy Dispersive X-ray Spectrometry, edited by K.F.J. Heinrich, D.E. Newbury, R.L. Myklebust, and C.E. Fiori, pp. 273-296.  National Bureau of Standards Special Publication 604, Washington, D.C.